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Physics of the frequency response of rectifying organicSchottky diodes

Stephane Altazin, Raphael Clerc, Romain Gwoziecki, Jean-Marie Verilhac,Damien Boudinet, Georges Pananakakis, Gérard Ghibaudo, Isabelle Chartier,

Romain Coppard

To cite this version:Stephane Altazin, Raphael Clerc, Romain Gwoziecki, Jean-Marie Verilhac, Damien Boudinet, et al..Physics of the frequency response of rectifying organic Schottky diodes. Journal of Applied Physics,American Institute of Physics, 2014, 115, pp.064509. 10.1063/1.4865739. hal-00947497

1

Physics of the Frequency Response

of Rectifying Organic Schottky Diodes

a,b

Stéphane Altazin*, cRaphaël Clerc,

aRomain Gwoziecki,

aJean-Marie Verilhac,

aDamien Boudinet,

bGeorges Pananakakis,

bGérard Ghibaudo,

aIsabelle Chartier

and aRomain Coppard

a CEA/LITEN/LCI, 38054 Grenoble (France)

b IMEP-LAHC, Minatec Grenoble INP, Grenoble, France.

* Now with Fluxim AG, Technoparkstrasse 2, CH-8406 Winterthur, Switzerland

c Laboratoire Hubert Curien (UMR 5516 CNRS), Université Jean Monnet, Institut d’Optique

Graduate School,18 rue Benoît Lauras 42000, Saint-Etienne, France.

Corresponding author : (raphael.clerc@institutoptique.fr)

Keywords: Organic Schottky diode, RFID Tag, rectifying circuit, Frequency, time dependent

response, modeling)

2

Abstract

The frequency response of unipolar organic Schottky diodes used in a rectifying circuit, such as

an RFID tag, has been investigated in detail. The time dependent response of rectifying circuits has

been simulated solving both the Drift Diffusion and Poisson equations to model the hole transport

within the diode, coupled with time dependent circuit equations. Several approximations have also

been discussed. It turns out that the cut off frequency of the rectifying circuit is indeed limited by

the carrier time-of-flight, and not by the diode equivalent capacitance. Simulations have also been

confirmed by comparison with experiments, involving diodes with different mobilities and

thicknesses. This work confirms that the 13.56 MHz frequency can be reached using polymer

semiconductors, as already experimentally demonstrated in the literature, by an adequate control of

the active layer thickness.

3

1. Introduction

One of the most promising applications of organic electronics consists in the production of low cost

radio frequency identification (RFID) tags on flexible substrates [1 - 3]. Several research groups

have already published encouraging results on organic devices, building blocks and complete

systems for RFID applications [1 - 15]. Among the most striking realizations, the world first “roll-to

-roll” organic RFID tag (a 64-bit tag working at a bit rate exceeding 100 b/s, readout by inductive

coupling at a base carrier frequency of 13.56 MHz) was presented in 2006 by Poly IC [1, 3]. More

recently, full organic RFID tags for barcode replacement, generating code sequences up to 128 bits,

have been demonstrated operating at the frequency communication frequency of 13.56 MHz [7 - 8].

In the latter case, the substrate, a PEN foil, was laminated on a rigid wafer and processed using

standard semiconductor equipment. In particular, the organic semiconductor (pentacene) was

evaporated in an ultra-high vacuum chamber.

One of the challenges of full organic RFID tags consists in achieving a rectifying diode operating at

frequency equal to or even higher than 13.56 MHz, using an organic semiconductor, materials

which are known to be penalized by their relatively poor transport properties. That is why the most

common procedure to manufacture these diodes is to evaporate small molecules semiconductors

(such as pentacene [4 - 10], copper phthalocyanine (CuPc) [11] or C60 [12]). Indeed, this technique

is known to be appropriate for growing high mobility polycrystalline layers (of the order of 0.15

cm2V

-1s

-1, such as for the pentacene used in [5]). To realize rectifying devices, these

semiconductors can then be introduced either in Schottky diodes [5 - 12], or diodes connected

organic transistors [13 – 14]. However, the evaporation deposition process may not be compatible

with the requirements of low cost roll-to-roll or printing technology.

Moreover, alternative solutions to evaporated small molecules, such as conjugate polymers, are

known to feature significantly lower mobilities (the best value being in the range of ~ 0.01 cm2V

-1s

-

1). In spite of that fact, there are several examples in the literature of successful polymer based

rectifying diodes operating at frequency higher than 13.56 MHz [1, 3, 4, 15 - 17]. The success of

4

these rectifying polymer diodes thus raises the following questions: is the mobility the only limiting

factor of the operating frequency of organic rectifying diodes? What are the device requirements

needed to achieve such frequency limit?

A literature search reveals that these questions have not been solved yet. The mechanisms limiting

the frequency operation of organic Schottky diodes in rectifying circuits do not appear to be well

understood, and in many aspects even controversial. Indeed, several articles attribute the time

limitation of rectifying circuits to the carrier time-of-flight within the diode [15, 17], but without

providing any convincing proof. That statement has been questioned in [5], proposing an improved

formula, presented as a “more stringent and realistic frequency limit”. Instead, other authors [11,

12] blame the diode equivalent capacitance, supposed to shunt the diode equivalent conductance at

high frequencies [11].

This controversial situation regarding the limiting frequency of organic Schottky diode is not so

surprising. Indeed, first of all, organic Schottky diodes essentially operate in a different way

compared to conventional inorganic junction, and are not completely understood yet. Moreover, the

rectification is intrinsically a highly non linear process, which cannot be described by the

conventional linear small signal approaches to analyze circuits. For those reasons, the frequency

response of those devices is a non trivial issue.

The aim of this work is to investigate in detail the physics of the organic Schottky diode frequency

response, and to determine the key parameters impacting it. First of all, our approach relies on an

accurate time dependent modeling of both the organic Schottky diode and the rectifying circuit, by

the means of numerical and analytical calculations. Theoretical results are then compared in details

with experiments, performed on home made printed diodes, operating at larger frequency than

13.56 MHz.

The present article is organized as follows: devices that have been fabricated and characterized are

presented in the first section. Then, three different approaches to model the organic diode within

5

non linear circuit background are presented and compared. Models and experiments are then

compared in the final section.

6

2. Device fabrication and characterization

In order to characterize the frequency operation of rectifying circuits, conventional organic

rectifying diodes have been fabricated, using a process compatible with the requirement of printed

electronics, and presented in the following section.

(a)

(b)

ITO / Pedot PSS electrode

TFB organic SC

Al electrode Thickness

∼ 70 – 430 nm

x axis

VoutVin

Organic diode

C R

(a)

(b)

ITO / Pedot PSS electrode

TFB organic SC

Al electrode Thickness

∼ 70 – 430 nm

x axis

ITO / Pedot PSS electrode

TFB organic SC

Al electrode Thickness

∼ 70 – 430 nm

x axis

VoutVin

Organic diode

C RVoutVin

Organic diode

C R

Figure 1. Schematic view of the diode used in this article (a) and schematic view of a rectifying

circuit studied in this article (b)

2.1. Organic diode process

Substrate is composed of a 100 nm layer of ITO deposited on glass, and then patterned. Then, a

50nm thin layer of gold is deposited to contact ITO with an external tips. PEDOT:PSS layer was

introduced in order to improve the hole injection. Then, the organic semiconductor (poly(fluorene-

alt-triarylamine) TFB in our case, see reference [18] for details), has been deposited by spin coating

at different rotation speeds, in order to get different thicknesses L. These thicknesses, measured by a

profilometer are 70nm, 260nm and 430nm. Then the device has been baked at 100°C during one

minute. 200 nm of Aluminium has been deposited through a shadow mask, to realise the top

contact. Finally, devices have been encapsulated by sticking, on its top, a glass plate. The substrate

used in the device measured here is composed of glass, but similar results can be obtained using

flexible substrate such as PEN. The schematic view of these diodes is presented in figure 1.

7

2.2. Electrical measurements

The static I-V characteristics of organic diodes have been measured by conventional high resolution

semiconductor analyser. Then, these devices have been introduced in the basic rectifying circuit

illustrated in figure 1. The capability of these diodes to perform a rectifying operation has been

demonstrated, as shown in figure 2, applying an input sinusoidal signal Vin, and measuring the

ouput signal Vout. If the rectifying operation has been successful, as it can been seen in figure 2,

performed at 15 MHz, the ouput voltage Vout should be nearly a constant signal.

-100 -50 0 50 100-10

-5

0

5

10

Vin

Vout

Experiments

Bia

s (V

)

Time (ns)

f=15 MHz t =70nm

-100 -50 0 50 100-10

-5

0

5

10

Vin

Vout

Experiments

Bia

s (V

)

Time (ns)

f=15 MHz t =70nm

Figure 2. Experimental Vin and Vout signals versus time, illustrating the successful operation of the

rectifying circuit at 15 MHz. This circuit is composed of a TFB diode of thickness 70 nm and area S

= 2 10-7

m2, a resistance of R=1 M, and a capacitor of C=1 µF.

8

3. Modeling of rectifying elementary circuit

In order to investigate the physics of the frequency operation of a Schottky diode in a rectifying

circuit, the operation of the full circuit, including the operation of the diode and the circuit, has to be

considered. Such a rigorous approach has been implemented and described in the first section. It

consists in a “mixed mode” approach, where Physics based equations are solved versus time, for

both the diode itself and the full circuit. Simplified approaches are also discussed in the following

sections. In particular, an original approach is proposed in the final section, saving computational

time, without any significant accuracy loss. Different approaches investigated in this work are

schematized in figure 3.

Approach 1 :Numerical solution of

Drift Diffusion

& Poisson equationsCd= S / t

Rd0(V) = 1 / Gd0(V)

Approach 2 :Quasi Static

equivalent circuit

where

d0 d0i (V) G (V) V= ⋅

is the diode static current d d d0 G G G

t

∂ −= −

∂ τ

Cd

Approach 3 :equivalent circuit with

time dependent Gd

where Gd(V,t) is given by the

phenomenological equation:

pdiv = 0

t

∂+ Φ

∂ p

D µ p Ex

∂Φ = − +

∂2

02

V e(p p )

x

∂= − −

ε∂

Rd(V,t) = 1 / Gd(V,t)

Approach 1 :Numerical solution of

Drift Diffusion

& Poisson equationsCd= S / t

Rd0(V) = 1 / Gd0(V)

Approach 2 :Quasi Static

equivalent circuit

where

d0 d0i (V) G (V) V= ⋅

is the diode static current d d d0 G G G

t

∂ −= −

∂ τ

Cd

Approach 3 :equivalent circuit with

time dependent Gd

where Gd(V,t) is given by the

phenomenological equation:

pdiv = 0

t

∂+ Φ

∂ p

D µ p Ex

∂Φ = − +

∂2

02

V e(p p )

x

∂= − −

ε∂

Rd(V,t) = 1 / Gd(V,t)

Figure 3. The three approaches to model the time dependent operation of the organic diode

considered in this work : 1/ time dependent drift diffusion equations coupled with Poisson; 2 /

elementary Gd, Cd0 static diode model; 3 / improved diode model including carrier response by a

phenomenological first order differential equation.

3.1. Time dependent mixed mode simulations

The intrinsically non linear nature of the rectifying process prevents to apply any standard small

signal approach, increasing the analysis level of complexity, even for the simple circuit of figure 1.

For this reason, the diode cannot be simulated alone, but together with the load resistance and

9

capacitance, in a time dependent approach. To this aim, the organic diode has been modeled using

the drift and diffusion formalism (in the time domain), using the theoretical background discussed

in [19]. Moreover, the time dependent drift and diffusion equation have been numerically solved,

according the procedure presented in [20]. The diode simulation is coupled with circuit transient

simulations, as illustrated in figure 3 (approach 1).

As the diode is planar (see figure 1), the transport of carriers has been modeled using a one

dimensional approach (i.e. along the transverse direction). Moreover, due to the nature of the

metallic contacts used in this work, these diodes are essentially unipolar devices, operating only

with holes (electrons transport can be ignored). The time dependent operation of the diode is

modeled by solving the following set of equations, composed of the continuity equation, coupled

with the Poisson equation:

pj pe 0

t x

∂∂+ =

∂ ∂ with: p p p

p Vj = e D e µ p

x x

∂ ∂− −

∂ ∂

(1)

where e is the elementary positive hole charge, p(x,t) the hole concentration, jp(x,t) the hole current

density, V(x,t) the electrostatic potential, µp (resp. Dp) the hole mobility (resp. diffusion

coefficient). For the sake of simplicity, the Einstein relation Dp = (kbT/e) µp, where kb is the

Boltzmann constant and T the temperature, is supposed to apply. The electrostatic potential V is

calculated solving self consistently the Poisson Equation (using a Newton algorithm) :

( )20

2

e p pV

x

−∂= −

∂

(2)

where represents the dielectric constant of the organic semiconductor, p0, the equilibrium hole

concentration.

The conventional “Schottky type” boundary conditions have been used, leading to :

1 a 2V(0) /e and V(L) V / e= ∆Φ = + ∆Φ (3)

10

1 20 0

b b

p(0) p exp and p(L) p expk T k T

∆Φ ∆Φ= − = −

(4)

where L represents the diode organic semiconductor thickness, Va the bias applied to the junction,

and 1 (resp. 2) the work function difference between the bottom (resp. top) electrode and the

organic semiconductor. Finally, the applied voltage Va is related to the input voltage Vin = V0

sin(ωt) and the output voltage Vout by the following set of equations :

a out inV (t) V (t) V (t)= −

(5)

out outd out in

d V VC i (V (t) V (t), t)

dt R+ = −

(6)

where C (resp. R) are the external capacitance (resp. resistance) of the rectifying circuit, and id = S

jp(L,t) the current flowing through the diode.

As the discrete time step ∆t has to be much lower than the input signal period 2π/ω, these

simulations may become excessively time-consuming in the high frequency regime. For this

reasons, a simplified model has been proposed in the following section.

3.2. Simplified model based on the equivalent circuit model Gd, Cd

In an attempt to understand better the rectifying circuit operation and to reduce computational time,

alternative approaches have been also investigated (represented by approaches 2 and 3 in Fig. 3).

The first simplified method (approach 2) consists in replacing the diode by a Gd0, Cd equivalent

circuit, where Gd0 is voltage dependent equivalent static conductance, and deduced from the static

id0(Va) characteristic and Cd represents the geometrical capacitance. This equivalent conductance

can be defined by :

d0 ad0 a

a

i (V )G (V )=

V

(7)

While this approach is in good agreement with approach 1 at low frequency, it however fails to

model the output signal around cut off, as seen in figure 4. This observation demonstrates that the

11

frequency response of the static diode capacitance cannot be simply captured by an equivalent

circuit model composed of a (time independent) capacitance Cd and conductance Gd0, as however

assumed in [11] and [12].

102

103

104

105

106

107

108

109

1010

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cd, Gd0

diode model(Approach 2)

t=100nm, Va=1V, VT=0.3VOu

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

Time dependent simulation (exact)

(Approach 1)

µ=0.01 cm2V-1s-1

R=100MΩ, C=1nF, S=0.1µm-2

Simulations

102

103

104

105

106

107

108

109

1010

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

102

103

104

105

106

107

108

109

1010

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cd, Gd0

diode model(Approach 2)

t=100nm, Va=1V, VT=0.3VOu

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

Time dependent simulation (exact)

(Approach 1)

µ=0.01 cm2V-1s-1

R=100MΩ, C=1nF, S=0.1µm-2

SimulationsR=100MΩ, C=1nF, S=0.1µm-2

Simulations

Figure 4 : Calculated average output voltage Vout(t) of the rectifying circuit versus input signal

frequency, using rigorous numerical simulation (approach 1) or simple diode model neglecting

carrier response transit time (approach 2). This figure shows that the approach 2, correct in the low

frequency regime, fails to capture the cut off at high frequency.

Actually, the fact that the geometrical capacitance of the diode (or by extension any kind of

capacitance) cannot cause the frequency cutoff could have been foreseen. Indeed, if we consider the

circuit labeled “approach 2” in figure 3, the following time dependent equation can be easily

deduced:

out in out outd0 in out d

d (V V ) d V Vi (V V ) C C

dt dt R

−− + = +

(8)

By integrating equation (8) over one full period of the input signal T = 2π/ω, we get:

t+T t+T

in out out outd0 d

t t

d (V V ) dV Vi (t) + C dt = C + dt

dt dt R

−

(9)

12

Once the stationary regime of the rectifying diode is achieved, Vout(t) becomes constant (Note that

this is also true in the high frequency regime, where the amplitude of Vout is almost zero). In this

case, both Vin and Vout are periodic signal, and thus:

( )t+T

in outin out in out

t

d (V V ) dt =V (t T) V (t T) V (t) V (t) 0

dt

− + − + − − =

(10)

In consequence, in the stationary regime, the output voltage Vout is given by the following equation:

t+T

out d0

t

RV = i dt

T

(11)

Equation (11) shows that, once stabilized, the output voltage Vout only depend on static parameters

(id0, R, T …). In particular, it becomes independent of the geometrical capacitance. Consequently,

the drastic drop of Vout observed in the high frequency regime cannot be induced by the diode

geometrical capacitance.

3.3. Simplified model based on the finite carrier response time

The diode time response appears rather limited by an intrinsic phenomenon, as shown by a closer

look at the time evolution of the hole concentration in the middle of the diode (see figure 5). At low

frequency, it oscillates from its lowest static value (corresponding to the static hole concentration

with a bias of –V0) to its highest static value (corresponding to a bias of +V0), following the

frequency of the input signal. Below the cut off frequency however, the hole concentration remains

constant, suggesting that the hole concentration fails to follow the input signal. Obviously, the hole

concentration is connected to the equivalent conductance, as detailed in the appendix 1, suggesting

that the time response of the hole concentration will also impact the equivalent conductance.

13

0 1 2 3 4 5 610

5

107

109

1011

1013

f=10GHz

f=10kHz f=100MHz

Ho

le C

on

cen

trati

on

(cm

-3)

Number of Periods

Simulations

0 1 2 3 4 5 610

5

107

109

1011

1013

f=10GHz

f=10kHz f=100MHz

Ho

le C

on

cen

trati

on

(cm

-3)

Number of Periods

Simulations

Figure 5: Carrier concentration versus time in the middle of the organic layer, at three input signal

frequencies, confirming the role of carrier response time in diode frequency behaviour.

(t=100nm, µ=10-2

cm2.V

-1.s

-1, ∆Φ1

=0 eV, ∆Φ2=0,42 eV, Na = 10

13cm

-3, Va=1V)

The previous approach 2 has thus been modified by introducing a characteristic response time to the

equivalent diode conductance Gd. Indeed, in the preceding case, the conductance Gd was supposed

to remain equal to the static conductance Gd0(V). In fact, this approximation implicitly assumes an

instantaneous response of the conductance to the applied signal Vin(t) – Vout(t). In the new

approach, the conductance has been modeled by a time dependent function Gd(V,t), responding to

the applied signal with a characteristic delay time τ. The dynamic of this response has been

empirically modeled in the simplest possible way, i.e. by the following first order differential

equation:

d d d0G (V) G (V) G (V)=

t

∂ −

∂

(12)

The next step consists in determining the characteristic delay time τ. As detailed in the second

appendix, analytical calculations suggest that a good approximation of this characteristic time

constant in this unipolar device is proportional to the carrier time-of-flight, given by :

14

2

p a T

L

µ (V V )τ = α

−.

(13)

Comparisons with numerical simulations of rectifying circuits (approach 1) suggest taking the

coefficient α equal to 0.82. The validity of this model (approach 3) will be investigated in details in

the next section.

15

4. Results and discussion

4.1. Comparison between the “mixed mode” simulations (approach 1) and simplified model

(approach 3)

103

104

105

106

107

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

fcfc

L=100nm, V0=1V, VT=0.3V

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

R=100MΩ, C=1nF, S=0.1µm-2

Simulations

µ=10-2 cm2V-1s-1

µ=10-5 cm2V-1s-1

104

105

106

107

108

0

1

2

3

4

5

fcfcV0=1V

V0=5V

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

L=100nm, µ=10-2 cm2V-1s-1, VT=0.3V

R=100MΩ, C=1nF, S=0.1µm-2

Simulations

104

105

106

107

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

fcfc

L=200nm

L=100nm

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

R=100MΩ, C=1nF, S=0.1µm-2

Simulations

µ=10-2 cm2V-1s-1, V0 = 1 V VT=0.3V

104

105

106

107

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

fc (VT=0.3V)

fc (VT=0.5V)

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

fc (VT=0.1V)

R=100MΩ, C=1nF, S=0.1µm-2

Simulations

µ=10-2 cm2V-1s-1

V0 = 1 V L = 100 nm

VT=0.1V

VT=0.3V

VT=0.5V

A B

C D10

310

410

510

610

70.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

fcfc

103

104

105

106

107

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

fcfc

L=100nm, V0=1V, VT=0.3V

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

R=100MΩ, C=1nF, S=0.1µm-2

SimulationsR=100MΩ, C=1nF, S=0.1µm-2

Simulations

µ=10-2 cm2V-1s-1

µ=10-5 cm2V-1s-1

104

105

106

107

108

0

1

2

3

4

5

fcfcV0=1V

V0=5V

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

L=100nm, µ=10-2 cm2V-1s-1, VT=0.3V

R=100MΩ, C=1nF, S=0.1µm-2

SimulationsR=100MΩ, C=1nF, S=0.1µm-2

Simulations

104

105

106

107

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

fcfc

L=200nm

L=100nm

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

R=100MΩ, C=1nF, S=0.1µm-2

SimulationsR=100MΩ, C=1nF, S=0.1µm-2

Simulations

µ=10-2 cm2V-1s-1, V0 = 1 V VT=0.3V

104

105

106

107

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

fc (VT=0.3V)

fc (VT=0.5V)

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

numericalanalytical

fc (VT=0.1V)

R=100MΩ, C=1nF, S=0.1µm-2

Simulations

µ=10-2 cm2V-1s-1

V0 = 1 V L = 100 nm

VT=0.1V

VT=0.3V

VT=0.5V

A B

C D

Figure 6. Calculated average output voltage Vout using approach 1 (full numerical solution) and

approach 3 (Equivalent circuit model with time dependant conductance Gd(Va, t) of the rectifying

circuit versus input signal frequency, for different parameters. Plot A : for two different

semiconductor thickness L = 100 nm and 200 nm; Plot B : for two different mobilities µ=10-2

cm2.V

-1.s

-1 and µ=10

-5 cm

2.V

-1.s

-1; Plot C : for two different amplitudes of Vin 1V and 5V, Plot D :

for three different threshold voltages 0.1 V, 0.3 V, 0.5 V. In absence of any other indications, the

parameters by defaults in the simulation are : L=100nm, p0= 3.31014

cm-3

, µ=10-2

cm2.V

-1.s

-1, Vin

amplitude =1V, and VT = 0.3 V.

16

The exact “mixed mode” approach 1 and the approximated approach have been extensively

compared, in order to investigate its validity. Simulation of the output voltage versus input signal

frequency have been performed, changing the semiconductor thickness L, the hole mobility µp, the

amplitude of the input signal V0 and the diode threshold voltage VT. Results have been plotted in

Figure 6. It turns out that the approximated simulations are in very good agreement with the full

numerical ones, in all the devices investigated. This observation confirms that: 1/ the Cd, Gd

equivalent circuit model fits very well the static operation of the diode in the low frequency range,

2/ the time-of-flight is very much representative of the diode characteristic response time τ, 3/ the

first order differential equation (12) provides a very good phenomenological description of the

equivalent conductance versus time. In the following, the approach 3, much more numerically

efficient, will be used to simulate the rectifying circuit.

4.2. Comparison with experiments

Simulations have been compared with experiments performed on the polymer diodes presented in

the first section. First of all, basic parameters such as mobility, equilibrium hole concentration, and

work functions have been extracted by comparison by fitting the static I-V characteristic of diodes

of several thicknesses, realized using the same process. At low voltages, to reproduce the static

current below threshold, it has been needed to account for shallow traps (i.e. taking into account the

trapping and de-trapping of holes in the Poisson equation, following the Lampert model [21]).

Interestingly, all the extracted parameters (see table 1) have been found, as expected, almost

independent of the diode thickness. Sample using a second organic polymer, the PCBTDPP

introduced in [22], has also been processed, in order to compare experiment and simulations one

sample featuring different mobilities.

Using parameters extracted in the static regime (table 1), as seen in figure 8, experimental output

voltage versus frequency curves have been nicely reproduced by simulations, without introducing

any other fitting parameters. This result confirms that the present model captures nicely the time

17

performance of devices featuring different thicknesses (figure 8) as well as different mobility

(figure 9). It thus confirms the validity of the “time-of-flight” formula used in the third approach,

and in particular its L2 and µ

-1 dependency. At last, as already seen in figure 2, note that

rectification at 13.56 MHz has been achieved using the thinnest diode (t = 70 nm), despite the

relatively low mobility 0.01 cm2V

-1s

-1 of the spin coated amorphous polymers TFB material.

-4 -3 -2 -110

-7

10-5

10-3

10-1

101

103 µ=0.01 cm2V-1s-1

: Experiment

: Simulation

t=70nmt=260nm

t=430nm

Cu

rren

t d

en

sit

y (

A.m

-2)

Applied Voltage (V)

Simulations & Experiments

-4 -3 -2 -110

-7

10-5

10-3

10-1

101

103 µ=0.01 cm2V-1s-1

: Experiment

: Simulation

t=70nmt=260nm

t=430nm

Cu

rren

t d

en

sit

y (

A.m

-2)

Applied Voltage (V)

Simulations & Experiments

Figure 7. Static I-V characteristic of the TFB organic diode in direct regime, and corresponding

simulations (using approach 1) (the reverse current, lower than the measurement set up resolution,

has not been shown).

Samples TFB

L=70nm

TFB

L=260nm

TFB

L=430nm

Trap

Concentration Nt (cm-3

)

1.4 1017

1.5 1016

1.0 1016

Doping

Concentration Na (cm-3

)

1012

1012

1012

Hole Concentration

at injecting contact (cm-3

)

6.9 1014

6.9 1014

6.9 1014

Mobility (cm2.V

-1.s

-1)

0.01

0.01

0.01

Work function

Difference between the

two contacts (eV)

0.97

0.97

0.97

Table 1. Parameters used to fit static experiments (figure 7). Mobility, equilibrium hole

concentration and work function has been found identical on the three diodes. Only the trap

concentration has been found to vary slightly from one sample to another.

18

105

106

107

0.01

0.1

1

10fc

fc fc

t=430nm

t=70nm

t=260nm

: Experiments

: Simulation

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

Simulations & Experiments

105

106

107

0.01

0.1

1

10fc

fc fc

t=430nm

t=70nm

t=260nm

: Experiments

: Simulation

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

Simulations & Experiments

Figure 8: Experimental and calculated average output voltage Vout versus input signal frequency,

for the three diodes, featuring various thicknesses L. Circuit is composed of a TFB diode of area S

= 2 10-7

m2, a resistance of R = 1 M, and a capacitor of C = 1 µF. The input signal amplitude is V0

= 6 V.

105

106

107

0.01

0.1

1

10

fc fc

µ=0.003 cm2V-1s-1 (PCBTDPP)

µ=0.01 cm2V-1s-1

: Experiments

: Simulation

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

(TFB)

Simulations & Experiments

105

106

107

0.01

0.1

1

10

fc fc

µ=0.003 cm2V-1s-1 (PCBTDPP)

µ=0.01 cm2V-1s-1

: Experiments

: Simulation

Ou

tpu

t V

olt

ag

e V

ou

t(V

)

Frequency (Hz)

(TFB)

Simulations & Experiments

Figure 9. Experimental and calculated average output voltage Vout versus input signal frequency,

for two diodes with different organic semiconductors (TFB and PCBTDPP). Circuit is composed of

a diode of area S = 2 10-7

m2, a resistance of R=1 M, and a capacitor of C=1 µF. The active layer

thickness is L = 260 nm. The input signal amplitude is V0 = 6 V.

19

4.3. Discussions

The validity of the “time-of-flight” formula for the cut off frequency has simple consequences,

illustrated in figure 10, where the cut off frequency has been plotted versus active layer thickness L,

for several mobilities µ. In this plot, the shadowed areas indicate either too low cut off frequency or

too thin organic thickness (the limit being arbitrary estimated at L > 40 nm). In order to exceed the

minimum frequency value of 13.56 MHz, low mobilities µ of polymer semiconductors, provided

that they exceed the minimum value of 10-4

cm2V

-1s

-1, can be simply compensated by using diodes

with ultra thin organic layer L, in order to minimize the ratio L2/µ. Some of the available

experimental data on the cut off frequency versus thicknesses have been also indicated in this

figure. These plots suggest that experimental data are consistent with organic mobilities in the 10-3

-

10-2

cm2V

-1s

-1 range, which is a reasonable conclusion. The reference [5] (fc > hundreds of

megahertz, L = 160 nm) is also in line with the “time-of-flight” theory, as the reported mobility in

this work was 0.15 cm2V

-1s

-1.

µ = 1 cm 2.V -1.s -1

µ = 1.10 -4cm2.V-1.s -1fc too low

10 100 1000

105

107

109

1011

Cu

toff

Fre

qu

en

cy

f c(H

z)

Diode Thickness (nm)

µ = 0.01 cm2.V -1.s -1

This workSpin coated TFB

L = 70 nm

[9]

µ = 1 cm 2.V -1.s -1

µ = 1.10 -4cm2.V-1.s -1

10 100 1000

105

107

109

1011

c

µ = 0.01 cm2.V -1.s -1

This workSpin coated TFB

L = 70 nm

[5]

µ = 0.01 cm2V-1s-1

[11]

L too small

(for printing)

[10]13.56 MHz

Simulations

µ = 1 cm 2.V -1.s -1

µ = 1.10 -4cm2.V-1.s -1fc too low

10 100 1000

105

107

109

1011

Cu

toff

Fre

qu

en

cy

f c(H

z)

Diode Thickness (nm)

µ = 0.01 cm2.V -1.s -1

This workSpin coated TFB

L = 70 nm

[9]

µ = 1 cm 2.V -1.s -1

µ = 1.10 -4cm2.V-1.s -1

10 100 1000

105

107

109

1011

c

µ = 0.01 cm2.V -1.s -1

This workSpin coated TFB

L = 70 nm

[5]

µ = 0.01 cm2V-1s-1

[11]

L too small

(for printing)

[10]13.56 MHz

Simulations

Figure 10. Rectifying circuit cut off frequency versus diode thickness for various organic layer

mobilities. Shadowed areas indicate either too low cut off frequency or too thin organic thickness

(very difficult to achieve using printing deposition techniques). Some of the literature performances

are also reported for comparison.

20

As already pointed out in other works such as [12] for instance, our simulations (not shown here)

confirm that a ratio between the forward and reverse currents of two orders of magnitude is

sufficient to achieve an efficient rectification. Interestingly, the time-of-flight formula even suggests

that it is not desirable to further increase this ratio. Indeed, higher ratio would require a higher

threshold voltage, which would degrade the cut off frequency fc = µ (VA − VT )/ L2.

Finally, note that the measurement of the cut off frequency is also an easy way to extract the carrier

mobility. This method allows a measurement of the diode equivalent conductance cut off frequency

independently of the equivalent capacitance, which can be very useful. Indeed, a direct

measurement of the diode conductance by measuring its admittance Y, given by :

d a d aY G ( , V ) i C ( , V )= ω + ω ω . (14)

requires that the equivalent conductance Gd is not negligible compare to ωCd. This condition, which

may be difficult to achieve at high frequency, is however not needed in the rectifying circuit

configuration.

5. Conclusions

The frequency response of unipolar organic Schottky diodes used in a rectifying circuit, such as the

one needed in RFID tag for instance, has been investigated in details.

First of all, the time dependent response of rectifying circuits has been simulated solving both the

Drift Diffusion and Poisson equations to model the hole transport within the diode, together with

the time dependent circuit equations. Several schemes of approximation have also been considered.

The first one consists in replacing the diode by its static, voltage dependent, Gd0, Cd equivalent

circuit. Valid in the low frequency regime, this approach however fails to capture the performance

degradation obtained at high frequency, showing that the diode equivalent capacitance is not

responsible for the rectifying circuit cut off effect. This approach can however be improved by

introducing a time dependency to the equivalent conductance, by the means of a first order linear

differential equation. The critical delay of the conductance has been found reasonably well captured

21

by the carrier time-of-flight, as suggested by analytical calculation presented in appendix 2 and

confirmed by extensive comparison with numerical simulations.

Simulations have also been validated by comparison with experiments, featuring diodes with

different mobilities and thicknesses.

Finally, this work explains why the 13.56 MHz frequency can be achieved using polymer organic

semiconductors, as already experimentally demonstrated in the literature, compensating the poor

value of polymer mobility by the ultra thin thickness of the diode active layer.

6. Appendix 1 : equivalent conductance Gd of the organic diode

In this appendix, a simple expression of the equivalent conductance model for the organic diode is

derived. It can be used 1/ to derive the equivalent conductance in the static regime, 2/ to derive

approximated time dependent conductance.

Starting from the drift and diffusion current conservation equation:

p

p

j kT 1 p V=

e µ p e p x x

∂ ∂− −

∂ ∂

(15)

We take the integral on the full semiconductor layer, leading to:

L Lp

p0 0

j kT 1 pdx= dx (V(0) V(L))

e µ p e p x

∂− − −

∂ (16)

The previous equation can be simplified as :

Lp

a 2 1

p0

j kT p(L)dx= ln V ( ) / e

e µ p e p(0)− + − ∆Φ − ∆Φ

(17)

The boundary conditions on the hole concentration imply:

2 1

b

p(L)exp

p(0) k T

∆Φ − ∆Φ= −

(18)

In consequences, current density and applied voltage are connected by the following equation :

22

L

p

a

p0

j (x, t)dx = V

e µ p(x,t) (19)

In the static regime, the current density jp is uniform (current conservation), and thus the static

conductance is exactly given by:

L

p

d0

a p0

j S 1G S dx

V e µ p(x)= = ×

(20)

where S is the diode area. Interestingly, although this equation seems to account only for drift and

not diffusion, it includes however, as previously proved, both mechanisms (providing to use the

exact p(x) profile). Moreover, note that Gd0 depends on the applied voltage Va through p(x).

By extension of this previous static equation, in the next appendix, the time evolution of the

conductance will be approximated by the following equation:

L

d a

p0

1G (V , t) S dx

e µ p(x,t)≈ ×

(21)

7. Appendix 2 : Majority carrier response time

In the following appendix, a procedure to calculate the majority carrier response time within the

organic diode has been proposed. To this aim, the following simplifying assumptions have been

made. First of all, the electric field E has been assumed constant in the diode. This approximation is

strictly valid only in ultra thin low doped semiconductor layers.

a TV VE=

L

−

(22)

where Va represents the applied bias on the diode and VT, an empirical threshold voltage. In

absence of traps, this threshold voltage is simply given by the built in potential (∆Φ2 − ∆Φ1)/e. In

all case, it can be extracted using the semilogarithmic plot of the static I-V curve, and its value is

equal to the voltage at which the I-V curve, in the forward regime, is not an exponential anymore.

23

We found the threshold voltage to be equal to 3 V, 4 V and 5 V for the devices thicknesses of 70

nm, 260 nm and 430 nm respectively.

The value of this characteristic time can then be approximately estimated by solving the continuity

equation when a Heaviside signal is applied to the diode at t = 0. The diffusion mechanism,

essentially operating around Va ∼ VT, has been neglected for sake of simplicity. The continuity

equation then becomes:

p

p p + µ E 0

t x

∂ ∂=

∂ ∂

(23)

To solve this equation, the unilateral Laplace transform P(x, s) of the carrier concentration p(x, t)

has been used, leading to:

p

PsP p(x,0) + µ E = 0

x

∂−

∂

(24)

At t = 0, the diode was supposed to be at thermal equilibrium, with no applied voltage. In

consequences, the initial hole profile p(x, t = 0) is given by:

2 1c0

e ( )p(x, t=0) = p exp x

kT L

− × −

(25)

where pc0 is the fixed boundary condition given by equation (4). The solution of (24) is:

s xP(x,s) = A exp g(x)

µE

− +

(26)

where A is an integration constant and g a particular solution, given by:

c0 2 1

2 1

p e ( )g(x) exp x

e ( ) kT Ls µE

kT L

− = − − −

(27)

The application of the boundary condition at x=0 leads to:

stc0 c0

2 1 0

p pP(x=0, s) = A+ = e p(x=0,t) dt =

e ( ) ss µE

kT L

∞

−

−−

(28)

The later equation can be used to find the integration constant A, leading to:

24

1

1 2 1c0

e ( )A = p s s µE

kT L

−

− −

− −

(29)

Finally P can be written as follows:

c0 c0 2 1

2 1

p p e ( )s x s xP(x,s)= exp exp x exp

e ( )s µE k TL µEs µE

k TL

−

− + − − − − −

(30)

In order to calculate the inverse Laplace transform of P, we have used the following usual formula:

exp( s)(t > )

s

−→

(31)

1exp( t) (t > 0)

s + → −

(32)

n

n+1

exp( s) (t )exp( a(t )) (t > )

(s+ a) n!

− −→ − −

(33)

And the inverse Laplace Transform of p(x,s) can then be calculated:

2 1c0 c0

e ( ) x xp(x,t) = p exp (L µEt) t 0 p t 0

kT L µE µE

− × − × − < + × − >

(34)

Knowing the hole concentration, the conductance Gd can then be approximated by (see appendix 1

for details):

L

d a

p0

1G (V , t) S dx

e µ p(x,t)≈ ×

(35)

After calculations we find:

c0d

2 1

2 1

S e µ pG =

e( )kTL exp (L µEt) 1

kTL L LµE t + t < + L t >

e( ) µE µE

− − − × ×

−

(36)

We can see in that the expression of the diode conductance will change after a time H equal to:

25

2

H

a T

L L

µE µ (V V )τ = =

−.

(37)

26

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